I'm used to using "giving a damn" in a negative sense. Merriam-Webster defines the phrase as meaning:
give a damn: to care at all about someone or something —used in negative statements —.
So, easy examples include:
-
He doesn't give a damn what people think about him.
-
Nobody gives a damn about us.
-
I don't want to hear about her problems. I just don't give a damn.
-
They don't seem to care a damn about their future.
The following use of the phrase (widespread in the US I think) is confusing to my English ears. The context suggests (to me) that she doesn't care, but the affirmative usage suggests (at least to me) that she does?
She was middle-school teacher, and her husband was a firefighter. She could give a damn about Edmund Burke and the New York Post. She was not a crunchy con, and she found her brother annoying.
— Rod Dreather's Monastic Vision by Joshua Rothman for the May 1st 2017 issue of The New Yorker, p. 46
Best Answer
"give a damn" is defined as follows in the Cambridge Idioms Dictionary, 2nd edition:
Some of the other comments call the use of the phrase "could give a damn" to mean "does not give a damn" incorrect and have made analogy to "I couldn't care less" ("damn" might also be replaced by "rat's ass", "shit", or "fuck", incidentally). Logically, I don't think this phrase is truly analogous. It works because "could" (in the present) points to counterfactual situations -- consider the difference between "I can speak Spanish" and "I could speak Spanish if I would have paid more attention in Mr. Garcia's class." † Or consider the difference between answering an invitation "Yes, I can go skiing on Saturday" and "I could go skiing on Saturday, but that place is too expensive" -- you can't really switch "can" and "could" in these examples.
With that said, I think it would be helpful to pull up some actual examples to prove that this phrase is far and away most commonly used to imply the negative (i.e., to not care). I just searched Google News; the examples seem to skew toward colorful and opinionated rants. That's not too surprising considering the phrase, so I hope you won't read too much into the choice of stories to take these from.
Talking Points Memo, "Fox News Host: Obama ‘Could Give A Shit’ About The Threat Of Terrorism"
Real Clear Politics, "Bobby Knight at Trump Rally: I Could Give a Damn About Republicans or Democrats, 'At Heart He's A Great American'"
Philly Voice, "Miami's 'El Clasico' a blatant cash grab and everything that's wrong with American soccer"
The Spectator, "When Liberal Doves Become Russia Hawks"
Vice, "Shirley Manson Asked Me Out for Sushi for Some Reason"
By contrast, I could not find any examples of the phrase meaning "to care."
† "had paid" is the more standard way to say this but it seems stilted to me in this example.